A Generalization of the Savage-Dickey Density Ratio for Testing Equality and Order Constrained Hypotheses

TL;DR

This paper extends the Savage-Dickey density ratio to handle hypotheses with both equality and order constraints, enabling more flexible Bayesian hypothesis testing with computational simplicity.

Contribution

It introduces a generalized method for computing Bayes factors under complex constrained hypotheses, broadening the applicability of the Savage-Dickey ratio.

Findings

Extended the Savage-Dickey ratio to order constraints.

Applied the method to multivariate t tests and multinomial models.

Demonstrated computational advantages in constrained hypothesis testing.

Abstract

The Savage-Dickey density ratio is a specific expression of the Bayes factor when testing a precise (equality constrained) hypothesis against an unrestricted alternative. The expression greatly simplifies the computation of the Bayes factor at the cost of assuming a specific form of the prior under the precise hypothesis as a function of the unrestricted prior. A generalization was proposed by Verdinelli and Wasserman (1995) such that the priors can be freely specified under both hypotheses while keeping the computational advantage. This paper presents an extension of this generalization when the hypothesis has equality as well as order constraints on the parameters of interest. The methodology is used for a constrained multivariate t test using the JZS Bayes factor and a constrained hypothesis test under the multinomial model.